sciencefandomcom_el-20200214-history
Μηχανική Συνεχούς Μέσου
Μηχανική Συνεχούς Μέσου Continuum Mechanics, Μηχανική Συνεχών Μέσων thumb|300px| [[Φυσική ---- Φυσικοί Γης Νόμοι Φυσικής Νόμοι Φυσικής Θεωρίες Φυσικής Πειράματα Φυσικής Παράδοξα Φυσικής ]] - Είναι ένας Επιστημονικός Κλάδος της Φυσικής. - Η Μηχανική του συνεχούς μέσου άλλωστε περιλαμβάνει τη Ελαστομηχανική (Μηχανική παραμορφωσίμων σωμάτων) και τη Ρευστομηχανική (Μηχανική των Ρευστών). Ετυμολογία Η ονομασία "Μηχανική συνεχούς μέσου" σχετίζεται ετυμολογικά με την λέξη "Συνεχές Μέσο". Εισαγωγή For a more general derivation using tensors, we consider a moving body (see Figure), assumed as a continuum, occupying a volume V'', at a time ''t, having a surface area S'', with defined traction or surface forces per unit area represented by the stress vector \scriptstyle T_i^{(n)}\, acting on every point of every body surface (external and internal), body forces ''Fi per unit of volume on every point within the volume V'', and a velocity field ''vi, prescribed throughout the body. Following the previous equation, the linear momentum of the system is: : \int_S T_i^{(n)}dS + \int_V F_i dV = \frac{d}{dt}\int_V \rho\, v_i\, dV\,. By definition the stress vector is defined as \scriptstyle T_i^{(n)} \equiv \sigma_{ij}n_j\, , then : \int_S \sigma_{ij}n_j\, dS + \int_V F_i\, dV = \frac{d}{dt}\int_V \rho\, v_i\, dV\,. Using the Gauss's divergence theorem to convert a surface integral to a volume integral gives (we denote \scriptstyle \partial_j \equiv \frac{\partial}{\partial x_j}\, as the differential operator): : \int_V \partial_j\sigma_{ij}\, dV + \int_V F_i\, dV = \frac{d}{dt}\int_V \rho\,v_i\, dV\,. Now we only need to take care of the right side of the equation. We have to be careful, since we cannot just take the differential operator under the integral. This is because while the motion of the continuum body is taking place (the body is not necessarily solid), the volume we are integrating on can change with time too. So the above integral will be: : \frac{d}{dt}\int \rho\,v_i\, dV=\int \frac{\partial (\rho v_i)}{\partial t}\, dV +\oint \rho v_i v_k n_k dA\,. Performing the differentiation in the first part, and applying the divergence theorem on the second part we obtain: : \frac{d}{dt}\int \rho\,v_i\, dV =\int \left[ \left(\rho\frac{\partial v_i}{\partial t}+v_i\frac{\partial \rho}{\partial t}\right)+\partial_k (\rho v_i v_k)\right]\, dV\,. Now the second term inside the integral is: \partial_k (\rho v_i v_k)=\rho v_k \cdot \partial_k v_i +v_i\partial_k(\rho v_k)\,. Plugging this into the previous equation, and rearranging the terms, we get: : \frac{d}{dt}\int \rho\,v_i\, dV=\int\rho\leftt}+v_k\partial_k\rightv_i\,dV +\int\leftt}+\partial_k(\rho v_k)\rightv_i\,dV\,. We can easily recognize the two integral terms in the above equation. The first integral contains the convective derivative of the velocity vector, and the second integral contains the change and flow of mass in time. Now lets assume that there are no sinks and sources in the system, that is mass is conserved, so this term is zero. Hence we obtain: : \frac{d}{dt}\int \rho\,v_i\, dV=\int \rho\,\frac{Dv_i}{Dt}\, dV\, putting this back into the original equation: : \int_V \left[ \partial_j\sigma_{ij} + F_i - \rho \frac{D v_i}{Dt}\right]\, dV = 0\,. For an arbitrary volume the integrand itself must be zero, and we have the Cauchy's equation of motion : \partial_j\sigma_{ij} + F_i = \rho \frac{D v_i}{Dt}\,. As we see the only extra assumption we made is that the system doesn't contain any mass sources or sinks, which means that mass is conserved. So this equation is valid for the motion of any continuum, even for that of fluids. If we are examining elastic continua only then the second term of the convective derivative operator can be neglected, and we are left with the usual time derivative, of the velocity field. If a system is in equilibrium, the change in momentum with respect to time is equal to 0, as there is no acceleration : \sum{\mathbf{F}} = {\mathrm{d}\mathbf{p} \over \mathrm{d}t}=\ m\mathbf{a}_{cm}= 0\, or using tensors, : \partial_j\sigma_{ij} + F_i = 0\,. These are the equilibrium equations which are used in solid mechanics for solving problems of linear elasticity. In engineering notation, the equilibrium equations are expressed in Cartesian coordinates as : \frac{\partial \sigma_x}{\partial x} + \frac{\partial \tau_{yx}}{\partial y} + \frac{\partial \tau_{zx}}{\partial z} + F_x = 0\, : \frac{\partial \tau_{xy}}{\partial x} + \frac{\partial \sigma_y}{\partial y} + \frac{\partial \tau_{zy}}{\partial z} + F_y = 0\, : \frac{\partial \tau_{xz}}{\partial x} + \frac{\partial \tau_{yz}}{\partial y} + \frac{\partial \sigma_z}{\partial z} + F_z = 0\,. Υποσημειώσεις Εσωτερική Αρθρογραφία *Κλασσική Μηχανική *Μηχανική Βιβλιογραφία # E. Becker und W. Bürger, Kontinuumsmechanik, Teubner, 1975 # R. L. Bisplinghoff, J. W. Mar and T. H. H. Pian, Statics of Deformable Solids, Dover, 1965. # P. Chadwick, Continuum Mechanics, Dover, 1976. # W.F. Chen and D. J. Han, Plasticity for Structural Engineers, Springer, 1988. # D.C. Kay, Tensor Calculus, Schaum’s Outline Series,1988 # L.E. Malvern, Introduction to the Mechanics of Continuous Medium, Prentice-Hall, 1969. # Sommerfeld, Mechanik der deformierbaren Medien, Bd. II, Verlag Herri Deutsch, 1992. # A.J.M. Spencer, Continuum Mechanics, Dover, 1980 Ιστογραφία *Ομώνυμο άρθρο στην Βικιπαίδεια *Ομώνυμο άρθρο στην Livepedia *[ ] *[ ] Category: Φυσική Category: Επιστήμες Category: Μηχανική